[Continuity Problem] Is f(x) which is continuous at x=a necessarily...

First thanks for taking time to view my question.

Is f(x) which is continuous at x=A necessarily continuous at a certain neighborhood of A?

I am quite confused since many people say yes. But if f(x) is also continuous at say, B, which is in the neighborhood of A, then this process would be infinite, and so f(x) should be continuous everywhere.